This sort of problem is usually heat-related but for an MX440, I'm sure that isn't the case. It's common for old cards to go off and give problems like those. Try the 9800PRO or, if u can afford it, the X800GTO. It is the best choice for AGP as far as power and price go.

the minimum requirement for NFS:MW, graphics wise is a nVidia GeForce 2 MX/GTS or greater with 32mb or greater so you should be ok. they are very good cards for there age but my guess would be a heat related problem as well.

@dazza-mate, the 9800PRO is a good, cheap card that will run all those games very easily. Also, the GeForce 6200 is a good alternative card if u prefer NVIDIA.
@N3051M, it's an MX440, so how would it tax the PSU in any way? All of it points to a bad card.